LAGRANGE_ND
Multivariate Lagrange Interpolation
LAGRANGE_ND
is a FORTRAN90 library which
is given a set of ND points X(*) in Ddimensional space, and constructs
a family of ND Lagrange polynomials P(*)(X), associating polynomial P(i)
with point X(i), such that, for 1 <= i <= ND,
P(i)(X(i)) = 1
but, if i =/= j
P(i)(X(j)) = 0
The library currently includes the following primary routines:

LAGRANGE_COMPLETE requires that the number of data points
ND is exactly equal to R, the number of monomials in D dimensions
of total degree N or less;

LAGRANGE_COMPLETE2, a version of LAGRANGE_COMPLETE with
improved "pivoting";

LAGRANGE_PARTIAL allows the number of data points
ND to be less than or equal to R, the number of monomials
in D dimensions of total degree N or less;

LAGRANGE_PARTIAL2, a version of LAGRANGE_PARTIAL
with improved "pivoting".
The set of ND polynomials P(*)(X) are returned as a set of three arrays:

PO(i) contains the order, the number of nonzero coefficients,
for polynomial i;

PC(i,j) contains the coefficient of the jth term in
polynomial i;

PE(i,j) contains a code for the exponents of the monomial
associated with the jth term in polynomial i.
Each value of PE(i,j) is an exponent codes which can be converted
to a vector of exponents that define a monomial. For example,
if we are working in spatial dimension D=3, then if PE(i,j)=13,
the corresponding exponent vector is (0,2,1), so
this means that the jth term in polynomial i is
PC(i,j) * x^0 y^2 z^1
An exponent code can be converted to an exponent vector by calling
mono_unrank_grlex().
Licensing:
The computer code and data files described and made available on this web page
are distributed under
the GNU LGPL license.
Languages:
LAGRANGE_ND is available in
a C++ version and
a FORTRAN90 version and
a MATLAB version.
Related Data and Programs:
LAGRANGE_INTERP_ND,
a FORTRAN90 library which
defines and evaluates the Lagrange polynomial p(x)
which interpolates a set of multivariate data, so that p(x(i)) = y(i).
SPARSE_INTERP_ND
a FORTRAN90 library which
can be used to define a sparse interpolant to a function f(x) of a
multidimensional argument.
TEST_INTERP_ND,
a FORTRAN90 library which
defines test problems for interpolation of data z(x),
depending on an Mdimensional argument.
Reference:

Philip Davis,
Interpolation and Approximation,
Dover, 1975,
ISBN: 0486624951,
LC: QA221.D33

Tomas Sauer, Yuan Xu,
On multivariate Lagrange interpolation,
Mathematics of Computation,
Volume 64, Number 211, July 1995, pages 11471170.
Source Code:
Examples and Tests:
List of Routines:

I4_CHOOSE computes the binomial coefficient C(N,K) as an I4.

I4MAT_PRINT prints an I4MAT.

I4MAT_PRINT_SOME prints some of an I4MAT.

I4VEC_CONCATENATE concatenates two I4VEC's.

I4VEC_PERMUTE permutes an I4VEC in place.

I4VEC_PRINT prints an I4VEC.

I4VEC_SORT_HEAP_INDEX_A does an indexed heap ascending sort of an I4VEC.

INTERPOLANT_VALUE evaluates a Lagrange interpolant.

LAGRANGE_COMPLETE: Complete Lagrange polynomial basis from data.

LAGRANGE_COMPLETE2: Complete Lagrange polynomial basis from data.

LAGRANGE_PARTIAL: Partial Lagrange polynomial basis from data.

LAGRANGE_PARTIAL2: Partial Lagrange polynomial basis from data.

LAGRANGE_PARTIAL3: Partial Lagrange polynomial basis from data.

LAGRANGE_PARTIAL4: Partial Lagrange polynomial basis from data.

MONO_BETWEEN_ENUM enumerates monomials in D dimensions of degrees in a range.

MONO_BETWEEN_NEXT_GRLEX: grlex next monomial, degree between N1 and N2.

MONO_NEXT_GRLEX returns the next monomial in grlex order.

MONO_TOTAL_ENUM enumerates monomials in D dimensions of degree equal to N.

MONO_TOTAL_NEXT_GRLEX: grlex next monomial with total degree equal to N.

MONO_UNRANK_GRLEX computes the monomial of given grlex rank.

MONO_UPTO_ENUM enumerates monomials in D dimensions of degree up to N.

MONO_VALUE evaluates a monomial.

PERM_CHECK checks that a vector represents a permutation.

POLYNOMIAL_AXPY adds a multiple of one polynomial to another.

POLYNOMIAL_COMPRESS compresses a polynomial.

POLYNOMIAL_PRINT prints a polynomial.

POLYNOMIAL_SORT sorts the information in a polynomial.

POLYNOMIAL_VALUE evaluates a polynomial.

R8_EPSILON returns the R8 roundoff unit.

R8_HUGE returns a very large R8.

R8COL_SEPARATION returns the "separation" of an R8COL.

R8MAT_IS_IDENTITY determines if an R8MAT is the identity.

R8MAT_PRINT prints an R8MAT.

R8MAT_PRINT_SOME prints some of an R8MAT.

R8MAT_TRANSPOSE_PRINT prints an R8MAT, transposed.

R8MAT_TRANSPOSE_PRINT_SOME prints some of an R8MAT, transposed.

R8VEC_CONCATENATE concatenates two R8VEC's.

R8VEC_PERMUTE permutes an R8VEC in place.

R8VEC_PRINT prints an R8VEC.

TIMESTAMP prints the current YMDHMS date as a time stamp.
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Last revised on 23 October 2014.